A physically meaningful parameter found in this model is a heat transfer coefficient from heating medium to product, which can be used to predict and control a pro[r]
Trang 1DATA-BASED MECHANISTIC MODELING APPROACH FOR PREDICTING THERMAL RESPONSE OF CONDUCTIVE FOOD DURING HEATING
PROCESSES
Nguyen Trung Truc1, To Quang Truong2, Le Thi Hoa Xuan3 and Vo Tan Thanh4
1 Vinh Long University of Technology Education, Vietnam
2 National Agro-Forestry-Fisheries Quality Assurance Department (zone 5)
3 Dong thap Community College, Vietnam
4 Department of Food Technology, Can Tho University, Vietnam
Received date: 29/07/2015
Accepted date: 19/02/2016 The application of data based mechanistic modeling approach to predict thermal histories of conductive foodstuffs during heating is reported In
the experiment, minced fish was filled in 307x113 steel cans as the con-ductive food Step increase in heating medium was applied while the product temperature was recorded The simplified refined instrumental variable algorithm was used as model parameter identification tool to obtain the best model order and parameters As a result, the first order transfer function model is proved to be sufficiently enough for describing the heat transfer from heating medium to product with a high statistical significance (R 2 > 0.99) In this model, a parameter related to the heat transfer coefficient was found and could be used to predict the product temperature during heating processes
KEYWORDS
Thermal processing, heat
transfer coefficient, modeling
Cited as: Truc, N.T., Truong, T.Q., Xuan, L.T.H., and Thanh, V.T., 2016 Data-based mechanistic modeling
approach for predicting thermal response of conductive food during heating processes Can Tho
University Journal of Science Vol 2: 63-68
1 INTRODUCTION
Thermal processing is one of the major
preserva-tion technologies used for producing safe and
shelf-stable food (Chen and Ramaswamy, 2004)
Temperature, which is the most important process
variable in most operations involving the
transfor-mation and preservation of foods, has a direct
in-fluence on the kinetics of chemical reactions, on
enzymatic and microbial activities, etc and it
changes with time of heating, and are broadly di-vided into two classes: “general method” and
“formula methods” The “general method” inte-grate the lethal effects by a graphical or numerical integration procedure based on the time-temperature data obtained from test containers pro-cessed under actual commercial processing condi-tions during “formula methods” make use of pa-rameters obtained from these heat penetration data together with several mathematical procedures to
Trang 2Most studies have only focused on the finding of
the best fitting of transfer function in thermal
pro-cessing (black box model), i.e the black box
mod-eling (Glavina et al., 2006; Ansorena and Scala,
2009; Ansorena et al., 2010) However, there is
still a need for finding a physically meaningful
parameter in a transfer function to predict and
con-trol product temperature during heating
The objective of this work is to characterize the
thermal response of canned conductive food in
batch retorts and try to model the process by using
a data-based mechanistic modeling approach
2 MATERIALS AND METHODS 2.1 Laboratory test equipment
In this experiment, minced fish was filled in 307x113 steel cans as the conductive food To ob-tain the temperature profiles, the calibrated type T thermocouples are positioned in the center of cans
as Figure 1 and all thermocouples were connected
to a digital data logger (Keithley 2700, USA)
(a) (b)
Fig 1: (a) Analog Keithley 2700; (b) Sensor position
During thermal treatment, the container was placed
in a water bath The product and water temperature
during experiment are recorded at 10-second
interval
To obtain the data sets for dynamic modeling, the
steps up of temperature was adjusted from 50 to
80oC, heating medium and product temperature
were monitored for 160 minutes as Figure 2
50
80
Product temperature Heating medium Setpoint temperature
o C)
Fig 2: Step up experiment
2.2 Data-based mechanistic (DBM) modeling approach
The term “data-based mechanistic modeling” was first introduced by Young and Lees in 1992 (cited
in Young, 2002) This modeling approach obtained initially from the analysis of observational time-series but was only considered credible if it can be interpreted in the physically meaningful terms As illustrated in Figure 3, the DBM approach consists
of data based and mechanistic phases The first step
in DBM modeling is to identify a suitable mathe-matical model from a generic model class that is both capable of explaining the data in a parametri-cally efficient manner and having minimal com-plexity in terms of model order and model parame-ters After this initial black-box modeling stage is complete, the model is interpreted in a physically meaningful, mechanistic manner based on the na-ture of the system under study and the physical, chemical, biological or socio-economic laws that are most likely to control its behavior
Trang 3Fig 3: Data based mechanistic (DBM) modeling technique
A continuous-time transfer function model for a
single-input single-output (SISO) system has the
following general form:
( )
( )
( )
B s
and ( )y t x t( )e t( ) or
( )
( )
B s
Where A(s) and B(s) are the following polynomials
in the derivative operator s d
dt
1
1
B s b s b s b s b
2.2.1 Data phase in thermal processing
High frequent data (10-second interval) was
ob-tained from dynamic experiments In the
“Data-based phase”, a dynamic transfer function model
was fitted through data and evaluated on its
accu-racy Although other techniques are available, in
The ability to estimate the parameters represents only one side of the model identification problem Equally important is the problem of objective
mod-el order identification resulting in low complexity The process of model order identification can be assisted by the use of well-chosen mathematical measures which indicate the presence of over pa-rameterization A reasonably successful identifica-tion procedure used to select the most appropriate model structure is based on the minimization of the
young identification criterion, YIC (Young et al.,
1981) The Young Identification Criterion (YIC) is
a heuristic statistical criterion, which consists of three elements The first term provides a normal-ized measure of how well the model explains the data: the smaller the variance of the model residu-als in relation to the variance of the measured out-put, the more negative this term becomes The sec-ond term is a normalized measure of how well the model parameter estimates are defined for the order model, the smaller the relative error variance, the better defined are the parameter estimate in statistic terms, and this is one more reflected in a more neg-ative value for the term The third term provides a
Trang 4and the standard error on this parameter estimates
becomes larger in relation to the estimated values
Consequently, the model, which minimizes the
YIC, provides a good compromise between
good-ness of fit and parametric efficiency While the
YIC ensures that the model is not over
parameter-ized, it is not always good at discriminating models
that have a lower order than the ‘best’ model
Be-cause of this, the YIC will often, if applied strictly,
identify a model that is under-parameterized
Therefore, it is used together with the coefficient of
determination R2 If the YIC identified model has
an adequate R2, which is not significantly lower
than the R2 of the higher order models, it may be
fully accepted as the best model in identification
terms
2.2.2 Mechanistic phase in thermal processing
Heating medium
Fig 4: Heat transfer during heat treatment
Assuming uniformity of product temperature
dur-ing heat treatment, the heat transfer between
heat-ing medium to product as shown in Figure 4 is
governed by the following equation:
m
d ( )
d
T t
Where m: mass of product (kg); Cp,m: specific heat
of product (J/kg oC); km: heat transfer coefficient
(W/m2 oC); Sm: surface of product (m2); Ti(t):
heat-ing medium temperature at time (oC); Tm(t):
prod-uct temperature at time (oC)
The Eq 1 can be rewritten as:
p,m
( ( ) ( ))
Let
p,m
k S
m C
Where α is a heat transfer rate (1/s)
The Eq (2) can be written as:
m
d ( )
d
T t
T t T t
Under steady state condition dT m 0
dt
, and Eq
(4) will become
(5)
If we only consider small temperature
perturba-tions (ti(t), tm(t)) around steady state, subtracting
Eq (5) from Eq (4) results in:
m
d ( )
d
t t
t t t t
After converting Eq (6) with the Laplace operator,
the transfer function results in:
s
The value in Eq 3 contains an important param-eter; it is a heat transfer coefficient km which is related to medium characteristics, medium velocity and surface of product
3 RESULTS AND DISCUSSION 3.1 Change of heating medium and product temperature
Heating medium and product temperature were recorded and performed in Figure 5 Product tem-perature was reached to medium temtem-perature after
100 min
Trang 50 100 200 300 400 500 600 700 800 900 45
50 55 60 65 70 75 80 85 90
Product temperature Medium temperature
Time (x 10s)
o C)
Fig 5: Heating medium and product temperature during experiment 3.2 Data-based phase in heating process
Applying continuous-time simplified refined
in-strumental variable (SRIV) algorithm (Young,
1981) to estimate the parameters in the first and
second order transfer function, based on coefficient
of determination R2 and minimization of the YIC value in the test as the example of for the calculat-ing method
Table 1: The model parameter estimates for heating process
First order [0, 1, 28] a1= 0.0105 0.0077 0.9999 -23.15
b0= 0.0106 Second order
[1, 2, 28] a1= 0.0191
0.0057 0.999 -12.52
a2= 0.0001
b0= 0.0104
b1= 0.0001
TF: transfer function; SE: standard error of equations; R 2 : coefficient of determination; YIC: Young identification crite-rion; m, n and , denominator, numerator and time delay; a 1 , a 2 , b 0 , b 1 , parameters in the first and second order of transfer function
50 60 70 80
o C)
Moi truong San pham Phong doan
-0.4 -0.2 0 0.2 0.4
Thoi gian (x 10 giay)
o C)
Bieu do sai so
Fig 6: The output of the first order of transfer function model compared with the measured
tempera-ture response (above) and residual plot (below)
with the measured temperature response and the
Trang 63.3 Physical meaningful parameter in a model
The second step in data-based mechanistic
model-ing approach is interpreted in a physically
mean-ingful way based on the nature of the system
(Young and Garnier, 2006) In this research, the
first order of fitted model was selected to seek the
physical meaning of this process From Eq (7) and
(8), the importantly found value is equal to b0,
was defined as a “heating rate” term in relation to
heat transfer coefficient from the heating medium
to product and the estimated parameters a1 and b0
are not very different, also proved the accuracy of
selected model
3.4 Applying of a selected model for predicting
of product temperature during heat treatment
The transfer function performed in Eq 8 contained
a physically meaning parameter So, it is possible
to apply to predict product temperature during heat treatment and can be used to online calculate
F-value (Least Sterilizing Value) during heat
treat-ment
The algorithm to predict product temperature was presented in Figure 7
Recording of
heating medium
temperature
Initial
temperature of
heating medium
Initial product temperature
T ref
z
F value
Product temperature
Predicting of product temperature
Online calculate F value
?
?
?
?
Fig 7: The algorithm to predict product temperature during heating
From Figure 7, with initial of product temperature,
initial temperature of heating medium and
record-ing heatrecord-ing medium, the predictrecord-ing product
tem-perature can be obtained It is a basic for online
calculating of the F-value with temperature
refer-ence (Tref) and thermal resistance (z)
4 CONCLUSIONS
The application of data-based mechanistic
model-ing approach to predict thermal histories of
con-ductive foodstuffs when surroundings present
dif-ferent forcing functions during heating of processes
was reported in this paper The comparison of
ex-perimental data with proposed model presented a
very satisfactory result with the first-order transfer
function A physically meaningful parameter found
in this model is a heat transfer coefficient from
heating medium to product, which can be used to
predict and control a product temperature during
heating process
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